TEICHMULLER SPACES AND GEOMETRY OF MOBIUS TRANSFORMATIONS

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My research during this term consists mainly of the following three branches :1. Consideration of the relation between angle parameters and the geometry of Mobius transformations.2. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.3. Characterization of simple dividing loops on Riemann surfaces analytically.In order to obtain global real analytic and simple representations of the Teichmuller spaces, I introduced new angle parameters. I showed that the Theichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces.Considering the axes of the generators and these products of Fuchsian groups, for example the once-holed torus Fuchsian groups, I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters, using the one-half powers of Mobius transformations an … More d the hyperbolic geometry. From these observations, the much relation and information of angle parameters were obtained. From such information of angle parameters, I tried to represent the Teichmuller modular groups by only angle parameters. I considered the following :(1) Relation between angle parameters and length parameters representing these groups.(2) Concrete description of such groups by only angle parameters in the cases that it is easy to calculate.(3) Choice of angle parameters (inductively) that simply represent the Theichmuller modular groups of the general cases.I especially studied the representation of the Theichmuller modular groups of a once-holed torus and a compact Riemann surface of genus 2.Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, using the lifts of a Fuchsian group G representing S to the special linear group SL (2, C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L.Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace. Less